Problem

For \(f(x)~=~(x~-~2)^2~+~3\) restricted to \(x~\ge~2\) and \(f^{-1}(x)~=~\sqrt{x~-~3}~+~2\), reflect the vertex and point \((3,~4)\), then state the original and inverse domains and ranges.

Neutral card showing f(x)=(x−2)²+3 restricted to x≥2 and f⁻¹(x)=√(x−3)+2, without reflected features, points, domains, or ranges.

Big Picture

What this problem is really about

Verify an inverse formula through several linked reflections rather than one algebraic resemblance. Swap the coordinates of the original anchor and a second point, check both images in the inverse rule, and then exchange the domain and range. Agreement across points, feature type, and allowable sets is stronger evidence than any single check alone.

Answer choices

Choose your path.

Preview only

Features: vertex (2,3) reflects to inverse endpoint (3,2), while point (3,4) remains (3,4). Sets: original domain [2,∞) and range [3,∞) swap to inverse domain [3,∞) and range [2,∞).

Features: vertex (2,3) reflects to inverse endpoint (3,2), and point (3,4) reflects to (4,3). Sets: original domain [2,∞) and range [3,∞) swap to inverse domain [3,∞) and range [2,∞).

Features: vertex (2,3) reflects to inverse endpoint (3,2), and point (3,4) reflects to (4,3). Sets: original domain [2,∞) and range [3,∞) are copied as inverse domain [2,∞) and range [3,∞).

Features: vertex (2,3) remains inverse endpoint (2,3), while point (3,4) reflects to (4,3). Sets: original domain [2,∞) and range [3,∞) remain the inverse domain and range without swapping.

The choices are visible. Checking your work and seeing which answer is correct happen inside Gozunta.

For \(f(x)~=~(x~-~2)^2~+~3\) restricted to \(x~\ge~2\) and \(f^{-1}(x)~=~\sqrt{x~-~3}~+~2\), reflect the vertex and point \((3,~4)\), then state the original and inverse domains and ranges.

What this problem is really about

Turn the preview into practice.

Gozunta lets you study this problem—and more than 10,000 others.